਀㰀琀椀琀氀攀㸀戀椀漀最爀愀瀀栀礀  愀渀搀 眀漀爀欀 圀椀氀栀攀氀洀 圀椀攀渀⸀ 吀栀攀 一漀戀攀氀 倀爀椀稀攀 椀渀 倀栀礀猀椀挀猀 ਀㰀䴀䔀吀䄀 栀琀琀瀀ⴀ攀焀甀椀瘀㴀䌀漀渀琀攀渀琀ⴀ匀琀礀氀攀ⴀ吀礀瀀攀 挀漀渀琀攀渀琀㴀琀攀砀琀⼀挀猀猀㸀㰀䰀䤀一䬀 栀爀攀昀㴀∀猀琀礀氀攀⸀挀猀猀∀ 琀礀瀀攀㴀琀攀砀琀⼀挀猀猀 爀攀氀㴀猀琀礀氀攀猀栀攀攀琀㸀 ਀㰀戀漀搀礀㸀 ਀ ਀㰀栀㈀ 愀氀椀最渀㴀挀攀渀琀攀爀㸀  圀椀氀栀攀氀洀 圀椀攀渀 The Nobel Prize in Physics 1910 ਀㰀栀㌀ 愀氀椀最渀㴀挀攀渀琀攀爀㸀戀椀漀最爀愀瀀栀礀 愀渀搀 眀漀爀欀㰀⼀栀㌀㸀㰀栀㐀㸀㰀椀洀最 愀氀椀最渀㴀氀攀昀琀 戀漀爀搀攀爀㴀㈀ 猀爀挀㴀∀眀椀攀渀⸀最椀昀∀㸀 㰀椀洀最 愀氀椀最渀㴀爀椀最栀琀 戀漀爀搀攀爀㴀㈀ 猀爀挀㴀∀眀椀攀渀⸀樀瀀最∀㸀 
Laws of radiation of heat਀䠀攀愀琀攀搀 䈀漀搀椀攀猀 刀愀搀椀愀琀攀 

਀圀攀 猀栀愀氀氀 渀漀眀 琀甀爀渀 琀漀 愀渀漀琀栀攀爀 瀀甀稀稀氀攀 挀漀渀昀爀漀渀琀椀渀最 瀀栀礀猀椀挀椀猀琀猀 愀琀 琀栀攀 琀甀爀渀 漀昀 琀栀攀 挀攀渀琀甀爀礀 ⴀⴀ 樀甀猀琀 栀漀眀 栀攀愀琀攀搀 戀漀搀椀攀猀 爀愀搀椀愀琀攀⸀ 吀栀攀爀攀 眀愀猀 愀 最攀渀攀爀愀氀 甀渀搀攀爀猀琀愀渀搀椀渀最 漀昀 琀栀攀 洀攀挀栀愀渀椀猀洀 椀渀瘀漀氀瘀攀搀 ⴀⴀ 栀攀愀琀 眀愀猀 欀渀漀眀渀 琀漀 挀愀甀猀攀 琀栀攀 洀漀氀攀挀甀氀攀猀 愀渀搀 愀琀漀洀猀 漀昀 愀 猀漀氀椀搀 琀漀 瘀椀戀爀愀琀攀Ⰰ 愀渀搀 椀琀 眀愀猀 欀渀漀眀渀 琀栀愀琀 琀栀攀 洀漀氀攀挀甀氀攀猀 愀渀搀 愀琀漀洀猀 眀攀爀攀 琀栀攀洀猀攀氀瘀攀猀 挀漀洀瀀氀椀挀愀琀攀搀 瀀愀琀琀攀爀渀猀 漀昀 攀氀攀挀琀爀椀挀愀氀 挀栀愀爀最攀猀⸀ 䘀爀漀洀 琀栀攀 攀砀瀀攀爀椀洀攀渀琀猀 漀昀 䠀攀爀琀稀 愀渀搀 漀琀栀攀爀猀 䴀愀砀眀攀氀氀✀猀 瀀爀攀搀椀挀琀椀漀渀猀 琀栀愀琀 漀猀挀椀氀氀愀琀椀渀最 挀栀愀爀最攀猀 攀洀椀琀琀攀搀 攀氀攀挀琀爀漀洀愀最渀攀琀椀挀 爀愀搀椀愀琀椀漀渀 栀愀搀 戀攀攀渀 挀漀渀昀椀爀洀攀搀Ⰰ 愀琀 氀攀愀猀琀 昀漀爀 猀椀洀瀀氀攀 愀渀琀攀渀渀愀猀⸀ 䤀琀 眀愀猀 欀渀漀眀渀 昀爀漀洀 䴀愀砀眀攀氀氀✀猀 攀焀甀愀琀椀漀渀猀 琀栀愀琀 琀栀椀猀 爀愀搀椀愀琀椀漀渀 琀爀愀瘀攀氀攀搀 愀琀 琀栀攀 猀瀀攀攀搀 漀昀 氀椀最栀琀 愀渀搀 昀爀漀洀 琀栀椀猀 椀琀 眀愀猀 爀攀愀氀椀稀攀搀 琀栀愀琀 氀椀最栀琀 椀琀猀攀氀昀Ⰰ 愀渀搀 琀栀攀 挀氀漀猀攀氀礀 爀攀氀愀琀攀搀 椀渀昀爀愀爀攀搀 栀攀愀琀 爀愀搀椀愀琀椀漀渀Ⰰ 眀攀爀攀 愀挀琀甀愀氀氀礀 攀氀攀挀琀爀漀洀愀最渀攀琀椀挀 眀愀瘀攀猀⸀ 吀栀攀 瀀椀挀琀甀爀攀Ⰰ 琀栀攀渀Ⰰ 眀愀猀 琀栀愀琀 眀栀攀渀 愀 戀漀搀礀 眀愀猀 栀攀愀琀攀搀Ⰰ 琀栀攀 挀漀渀猀攀焀甀攀渀琀 瘀椀戀爀愀琀椀漀渀猀 漀渀 愀 洀漀氀攀挀甀氀愀爀 愀渀搀 愀琀漀洀椀挀 猀挀愀氀攀 椀渀挀氀甀搀攀搀 猀漀洀攀 漀猀挀椀氀氀愀琀椀渀最 挀栀愀爀最攀猀⸀ 䤀昀 漀渀攀 愀猀猀甀洀攀搀 琀栀愀琀 䴀愀砀眀攀氀氀✀猀 琀栀攀漀爀礀 漀昀 攀氀攀挀琀爀漀洀愀最渀攀琀椀挀 爀愀搀椀愀琀椀漀渀Ⰰ 眀栀椀挀栀 眀漀爀欀攀搀 眀攀氀氀 椀渀 琀栀攀 洀愀挀爀漀猀挀漀瀀椀挀 眀漀爀氀搀Ⰰ 眀愀猀 愀氀猀漀 瘀愀氀椀搀 愀琀 琀栀攀 洀椀挀爀漀猀挀漀瀀椀挀 猀挀愀氀攀 ⠀琀攀渀琀栀猀 漀昀 渀愀渀漀洀攀琀攀爀猀⤀Ⰰ 琀栀攀渀 琀栀攀猀攀 漀猀挀椀氀氀愀琀椀渀最 挀栀愀爀最攀猀 眀漀甀氀搀 爀愀搀椀愀琀攀Ⰰ 瀀爀攀猀甀洀愀戀氀礀 最椀瘀椀渀最 漀昀昀 栀攀愀琀 愀渀搀 氀椀最栀琀⸀ 


How is Radiation Absorbed? ਀㰀⼀戀爀㸀㰀瀀㸀㰀䈀爀㸀 What is meant by the phrase "black body" radiation? The point is that radiation from heated bodies depends to some extent on the body heated. Let us back up momentarily and consider how differently different materials absorb radiation. Some, like glass, seem to absorb light hardly at all-the light goes through. For a shiny metallic surface, the light isn't absorbed either, it gets reflected. For a black material like soot, light and heat are almost completely absorbed, and the material gets warm. How can we understand these different behaviors in terms of light as an electromagnetic wave interacting with charges in the material, causing them to oscillate and absorb energy from the radiation? In the case of glass, evidently this doesn't happen, at least not much. A full understanding of how this works needs quantum mechanics, but the general idea is as follows. There are charges-electrons-in glass that are able to oscillate in response to an applied external oscillating electric field, but these charges are tightly bound to atoms, and only oscillate at certain frequencies. It happens that for ordinary glass none of these frequencies correspond to those of visible light, so there is no resonance with a light wave, and hence little energy absorbed. Glass is opaque at some frequencies outside the visible range (in general, both in the infrared and the ultraviolet). These are the frequencies at which the electrical charge distribution in the atoms or bonds can naturally oscillate.਀㰀⼀戀爀㸀㰀戀爀㸀 ਀䄀 瀀椀攀挀攀 漀昀 洀攀琀愀氀 栀愀猀 攀氀攀挀琀爀漀渀猀 昀爀攀攀 琀漀 洀漀瘀攀 琀栀爀漀甀最栀 琀栀攀 攀渀琀椀爀攀 猀漀氀椀搀⸀ 吀栀椀猀 椀猀 眀栀礀 洀攀琀愀氀猀 挀愀渀 挀漀渀搀甀挀琀 攀氀攀挀琀爀椀挀椀琀礀⸀ 䤀琀 椀猀 愀氀猀漀 眀栀礀 琀栀攀礀 愀爀攀 猀栀椀渀礀⸀ 吀栀攀猀攀 甀渀愀琀琀愀挀栀攀搀 攀氀攀挀琀爀漀渀猀 漀猀挀椀氀氀愀琀攀 琀漀最攀琀栀攀爀 眀椀琀栀 氀愀爀最攀 愀洀瀀氀椀琀甀搀攀 椀渀 爀攀猀瀀漀渀猀攀 琀漀 琀栀攀 攀氀攀挀琀爀椀挀愀氀 昀椀攀氀搀 漀昀 愀渀 椀渀挀漀洀椀渀最 氀椀最栀琀 眀愀瘀攀⸀ 吀栀攀礀 琀栀攀洀猀攀氀瘀攀猀 琀栀攀渀 爀愀搀椀愀琀攀 攀氀攀挀琀爀漀洀愀最渀攀琀椀挀愀氀氀礀Ⰰ 樀甀猀琀 氀椀欀攀 愀 挀甀爀爀攀渀琀 椀渀 愀渀 愀渀琀攀渀渀愀⸀ 吀栀椀猀 爀愀搀椀愀琀椀漀渀 昀爀漀洀 琀栀攀 漀猀挀椀氀氀愀琀椀渀最 攀氀攀挀琀爀漀渀猀 椀猀 琀栀攀 爀攀昀氀攀挀琀攀搀 氀椀最栀琀⸀ 䤀渀 琀栀椀猀 猀椀琀甀愀琀椀漀渀Ⰰ 氀椀琀琀氀攀 漀昀 琀栀攀 椀渀挀漀洀椀渀最 爀愀搀椀愀渀琀 攀渀攀爀最礀 椀猀 愀戀猀漀爀戀攀搀Ⰰ 椀琀 椀猀 樀甀猀琀 爀攀爀愀搀椀愀琀攀搀Ⰰ 琀栀愀琀 椀猀Ⰰ 爀攀昀氀攀挀琀攀搀⸀ 

਀ Soot, like a metal, will conduct an electric current, although not nearly so well. There are unattached electrons, which can move through the whole solid, but they keep bumping into things-they have a short mean free path. When they bump, they cause vibration, like a pinball machine, so they give up energy into heat. Although the electrons in soot have a short mean free path compared to that in a good metal, they move very freely compared with electrons in atoms, so they can accelerate and pick up energy from the electric field in the light wave. They are therefore effective intermediaries in transferring energy from the light wave into heat. ਀䄀戀猀漀爀瀀琀椀漀渀 愀渀搀 䔀洀椀猀猀椀漀渀 

਀䠀攀愀琀攀搀 戀漀搀椀攀猀 爀愀搀椀愀琀攀 戀礀 瀀爀漀挀攀猀猀攀猀 樀甀猀琀 氀椀欀攀 琀栀攀 愀戀猀漀爀瀀琀椀漀渀 搀攀猀挀爀椀戀攀搀 愀戀漀瘀攀 漀瀀攀爀愀琀椀渀最 椀渀 爀攀瘀攀爀猀攀⸀ 吀栀甀猀Ⰰ 昀漀爀 猀漀漀琀 栀攀愀琀 挀愀甀猀攀猀 琀栀攀 氀愀琀琀椀挀攀 琀漀 瘀椀戀爀愀琀攀 洀漀爀攀 瘀椀最漀爀漀甀猀氀礀Ⰰ 最椀瘀椀渀最 攀渀攀爀最礀 琀漀 琀栀攀 攀氀攀挀琀爀漀渀猀 ⠀椀洀愀最椀渀攀 琀栀攀洀 愀猀 戀愀氀氀猀 椀渀 愀 瀀椀渀戀愀氀氀 洀愀挀栀椀渀攀 眀椀琀栀 猀琀爀漀渀最氀礀 瘀椀戀爀愀琀椀渀最 戀愀爀爀椀攀爀猀Ⰰ 攀琀挀⸀⤀ 愀渀搀 猀椀渀挀攀 琀栀攀 攀氀攀挀琀爀漀渀猀 愀爀攀 挀栀愀爀最攀搀 琀栀攀礀 爀愀搀椀愀琀攀 愀眀愀礀 攀砀挀攀猀猀 欀椀渀攀琀椀挀 攀渀攀爀最礀⸀ 伀渀 琀栀攀 漀琀栀攀爀 栀愀渀搀Ⰰ 琀栀攀 攀氀攀挀琀爀漀渀猀 椀渀 愀 洀攀琀愀氀 栀愀瘀攀 瘀攀爀礀 氀漀渀最 洀攀愀渀 昀爀攀攀 瀀愀琀栀猀Ⰰ 琀栀攀 氀愀琀琀椀挀攀 瘀椀戀爀愀琀椀漀渀猀 愀昀昀攀挀琀 琀栀攀洀 洀甀挀栀 氀攀猀猀Ⰰ 猀漀 琀栀攀礀 愀爀攀 氀攀猀猀 攀昀昀攀挀琀椀瘀攀 椀渀 爀愀搀椀愀琀椀渀最 愀眀愀礀 栀攀愀琀⸀ 䤀琀 椀猀 攀瘀椀搀攀渀琀 昀爀漀洀 挀漀渀猀椀搀攀爀愀琀椀漀渀猀 氀椀欀攀 琀栀椀猀 琀栀愀琀 最漀漀搀 愀戀猀漀爀戀攀爀猀 漀昀 爀愀搀椀愀琀椀漀渀 愀爀攀 愀氀猀漀 最漀漀搀 攀洀椀琀琀攀爀猀⸀  ਀㰀⼀戀爀㸀㰀戀爀㸀 At sufficiently high temperatures, all bodies become good radiators. Items heated until they glow in a fire look much more similar than they do at room temperature. For a metal, this can be understood in terms of a shortening of the mean free path by the stronger vibrations of the lattice interfering with the electron's passage. ਀伀戀猀攀爀瘀椀渀最 琀栀攀 䈀氀愀挀欀 䈀漀搀礀 匀瀀攀挀琀爀甀洀 

਀䄀渀礀 戀漀搀礀 愀琀 愀渀礀 琀攀洀瀀攀爀愀琀甀爀攀 愀戀漀瘀攀 愀戀猀漀氀甀琀攀 稀攀爀漀 眀椀氀氀 爀愀搀椀愀琀攀 琀漀 猀漀洀攀 攀砀琀攀渀琀Ⰰ 戀甀琀 琀栀攀 椀渀琀攀渀猀椀琀礀 愀渀搀 昀爀攀焀甀攀渀挀礀 搀椀猀琀爀椀戀甀琀椀漀渀 漀昀 琀栀攀 爀愀搀椀愀琀椀漀渀 搀攀瀀攀渀搀猀 漀渀 琀栀攀 搀攀琀愀椀氀攀搀 猀琀爀甀挀琀甀爀攀 漀昀 琀栀攀 戀漀搀礀⸀ 吀漀 洀愀欀攀 愀渀礀 瀀爀漀最爀攀猀猀 椀渀 甀渀搀攀爀猀琀愀渀搀椀渀最 爀愀搀椀愀琀椀漀渀Ⰰ 眀攀 洀甀猀琀 猀瀀攀挀椀昀礀 琀栀攀 搀攀琀愀椀氀猀 漀昀 琀栀攀 戀漀搀礀 爀愀搀椀愀琀椀渀最⸀ 吀栀攀 猀椀洀瀀氀攀猀琀 瀀漀猀猀椀戀氀攀 洀漀搀攀氀 椀猀 琀漀 挀漀渀猀椀搀攀爀 愀 戀漀搀礀 眀栀椀挀栀 椀猀 愀 瀀攀爀昀攀挀琀 愀戀猀漀爀戀攀爀Ⰰ 愀渀搀 琀栀攀爀攀昀漀爀攀 琀栀攀 戀攀猀琀 瀀漀猀猀椀戀氀攀 攀洀椀琀琀攀爀⸀ 䘀漀爀 漀戀瘀椀漀甀猀 爀攀愀猀漀渀猀Ⰰ 琀栀椀猀 椀猀 挀愀氀氀攀搀 愀 ∀戀氀愀挀欀 戀漀搀礀∀⸀  ਀㰀⼀戀爀㸀㰀戀爀㸀 A good model absorber is a small hole in the side of a box. Radiation impinging on the hole from outside enters the box and is absorbed as it is scattered around inside, so only a tiny fraction is re-emitted. ਀

਀圀攀 猀栀愀氀氀Ⰰ 琀栀攀爀攀昀漀爀攀Ⰰ 琀愀欀攀 愀 挀甀戀椀挀愀氀 戀漀砀 ⠀愀渀 漀瘀攀渀Ⰰ 椀渀 漀琀栀攀爀 眀漀爀搀猀⤀ 眀椀琀栀 愀 猀洀愀氀氀 栀漀氀攀 椀渀 琀栀攀 猀椀搀攀 愀猀 漀甀爀 洀漀搀攀氀 琀漀 愀渀愀氀礀稀攀 攀洀椀猀猀椀漀渀 漀昀 爀愀搀椀愀琀椀漀渀⸀ 䘀漀爀琀甀渀愀琀攀氀礀Ⰰ 琀栀椀猀 椀猀 愀 猀礀猀琀攀洀 眀栀椀挀栀 挀愀渀 愀氀猀漀 戀攀 椀渀瘀攀猀琀椀最愀琀攀搀 攀砀瀀攀爀椀洀攀渀琀愀氀氀礀 昀愀椀爀氀礀 攀愀猀椀氀礀⸀ 䄀 戀攀愀洀 漀昀 爀愀搀椀愀琀椀漀渀 昀爀漀洀 琀栀攀 栀漀氀攀 椀渀 琀栀攀 漀瘀攀渀 椀猀 瀀愀猀猀攀搀 琀栀爀漀甀最栀 愀 搀椀昀昀爀愀挀琀椀漀渀 最爀愀琀椀渀最 愀渀搀 瀀爀漀樀攀挀琀攀搀 漀渀 琀漀 愀 猀挀爀攀攀渀Ⰰ 眀栀攀爀攀 椀琀 椀猀 猀攀瀀愀爀愀琀攀搀 漀甀琀 戀礀 眀愀瘀攀氀攀渀最琀栀 ⠀漀爀 攀焀甀椀瘀愀氀攀渀琀氀礀 昀爀攀焀甀攀渀挀礀⤀⸀ 䄀 搀攀琀攀挀琀漀爀 椀猀 洀漀瘀攀搀 甀瀀 愀渀搀 搀漀眀渀 愀氀漀渀最 琀栀攀 猀挀爀攀攀渀 琀漀 昀椀渀搀 栀漀眀 洀甀挀栀 爀愀搀椀愀渀琀 攀渀攀爀最礀 椀猀 戀攀椀渀最 攀洀椀琀琀攀搀 椀渀 攀愀挀栀 昀爀攀焀甀攀渀挀礀 爀愀渀最攀Ⰰ 猀愀礀 琀栀攀 攀渀攀爀最礀 戀攀琀眀攀攀渀 昀爀攀焀甀攀渀挀礀 昀 愀渀搀 昀 ⬀ 搀昀 椀猀 刀吀⠀昀⤀搀昀⸀ 

਀ We can then plot RT(f) against f for a given temperature T. It is found experimentally that for small f, RT(f) is proportional to f2, a parabolic shape, but for increasing f it falls below the parabola, peaks at fmax, then falls quite rapidly to zero as f increases beyond fmax.਀㰀⼀戀爀㸀㰀戀爀㸀 ਀㰀⼀瀀㸀 ਀ ਀



਀ Suppose we now double the absolute temperature - how does that affect RT(f)?਀㰀⼀戀爀㸀㰀戀爀㸀 ਀䘀漀爀 琀栀漀猀攀 氀漀眀 昀爀攀焀甀攀渀挀椀攀猀 眀栀攀爀攀 刀吀⠀昀⤀ 眀愀猀 瀀愀爀愀戀漀氀椀挀Ⰰ 搀漀甀戀氀椀渀最 琀栀攀 琀攀洀瀀攀爀愀琀甀爀攀 搀漀甀戀氀攀猀 琀栀攀 椀渀琀攀渀猀椀琀礀 漀昀 琀栀攀 爀愀搀椀愀琀椀漀渀⸀ 䠀漀眀攀瘀攀爀Ⰰ 椀琀 椀猀 昀漀甀渀搀 琀栀愀琀 愀琀 ㈀吀 琀栀攀 挀甀爀瘀攀 昀漀氀氀漀眀猀 琀栀攀 ⠀搀漀甀戀氀攀搀⤀ 瀀愀爀愀戀漀氀椀挀 瀀愀琀栀 洀甀挀栀 昀甀爀琀栀攀爀 ⴀ 椀渀 昀愀挀琀Ⰰ 琀眀椀挀攀 愀猀 昀愀爀㨀 昀洀愀砀⠀㈀吀⤀ 㴀 ㈀昀洀愀砀⠀吀⤀⸀  ਀㰀⼀戀爀㸀㰀戀爀㸀 The curve R2T(f), then, reaches eight times the height of RT(f). It also spreads over twice the lateral extent, so the area under the curve, corresponding to the total energy radiated, increases sixteenfold on doubling the temperature. This is Stefan's Law of Radiation: the power P radiated from one square meter of black surface at temperature T:਀

਀ ਀㰀⼀戀爀㸀㰀戀爀㸀 Note that the graph above plots the energy intensity in the oven. This energy is being pumped out of the opening at speed c. When allowance is made for the fact that an opening of unit area is effectively less than unit area for radiation coming in at an angle, and half the energy is in waves moving in directions away from the hole, the actual rate of emission of energy through a hole of unit area is equal to (1/4)c times the energy intensity inside. The blue curve in the graph above is for 10,000K. The energy intensity plotted on the y-axis is in joules/m3 per unit frequency range, and it is evident on glancing at the graph that integrating to give the total intensity one will find something around 8 joules/m3. To find the heat energy radiated per second per square meter of surface at 10,000K, we multiply this energy density by c/4, which gives around 600 Megawatts per square meter. ਀

਀吀栀攀 甀瀀眀愀爀搀 猀栀椀昀琀 椀渀 昀洀愀砀 眀椀琀栀 吀 椀猀 猀漀洀攀琀栀椀渀最 攀瘀攀爀礀漀渀攀 椀猀 昀愀洀椀氀椀愀爀 眀椀琀栀 ⴀ 眀栀攀渀 愀渀 椀爀漀渀 椀猀 栀攀愀琀攀搀 椀渀 愀 昀椀爀攀Ⰰ 琀栀攀 昀椀爀猀琀 瘀椀猀椀戀氀攀 爀愀搀椀愀琀椀漀渀 ⠀愀琀 愀爀漀甀渀搀 㤀　　䬀⤀ 椀猀 搀攀攀瀀 爀攀搀 ⴀ 琀栀攀 氀漀眀攀猀琀 昀爀攀焀甀攀渀挀礀 瘀椀猀椀戀氀攀 氀椀最栀琀⸀ 䘀甀爀琀栀攀爀 椀渀挀爀攀愀猀攀 椀渀 吀 挀愀甀猀攀猀 琀栀攀 挀漀氀漀爀 琀漀 挀栀愀渀最攀 琀漀 漀爀愀渀最攀 琀栀攀渀 礀攀氀氀漀眀Ⰰ 愀渀搀 眀栀椀琀攀 愀琀 愀 瘀攀爀礀 栀椀最栀 琀攀洀瀀攀爀愀琀甀爀攀Ⰰ 猀椀最渀椀昀礀椀渀最 琀栀愀琀 愀氀氀 琀栀攀 瘀椀猀椀戀氀攀 昀爀攀焀甀攀渀挀椀攀猀 愀爀攀 戀攀椀渀最 攀洀椀琀琀攀搀 爀漀甀最栀氀礀 攀焀甀愀氀氀礀⸀  ਀㰀⼀戀爀㸀㰀戀爀㸀 The change in fmax is linear in the absolute temperature; ਀

਀ This is called Wein's Displacement Law.਀

਀䘀漀爀 攀砀愀洀瀀氀攀Ⰰ 琀栀攀 猀甀渀✀猀 猀甀爀昀愀挀攀 琀攀洀瀀攀爀愀琀甀爀攀 椀猀 㔀㜀　　䬀Ⰰ 愀渀搀 愀琀 琀栀愀琀 琀攀洀瀀攀爀愀琀甀爀攀 洀甀挀栀 漀昀 琀栀攀 攀渀攀爀最礀 椀猀 爀愀搀椀愀琀攀搀 愀猀 瘀椀猀椀戀氀攀 氀椀最栀琀⸀ 吀栀椀猀 椀猀 渀漀 愀挀挀椀搀攀渀琀 ⴀ 攀瘀漀氀甀琀椀漀渀 栀愀猀 愀搀愀瀀琀攀搀 甀猀 琀漀 猀攀攀 洀漀猀琀 攀昀昀椀挀椀攀渀琀氀礀 椀渀 琀栀攀 氀椀最栀琀 洀漀猀琀 爀攀愀搀椀氀礀 愀瘀愀椀氀愀戀氀攀⸀

਀ This shift in the frequency at which radiant power is a maximum is very important for harnessing solar energy, such as in a greenhouse. We need glass which will allow the solar radiation in, but not let the heat radiation out. This is feasible because the two radiations are in very different frequency ranges - 5700K and, say, 300K - and there are materials transparent to light but opaque to infrared radiation. This is only possible because fmax varies with temperature. ਀唀渀搀攀爀猀琀愀渀搀椀渀最 琀栀攀 䈀氀愀挀欀 䈀漀搀礀 匀瀀攀挀琀爀甀洀㨀 琀栀攀 䄀渀愀氀漀最礀 眀椀琀栀 愀 䜀愀猀 

਀䔀砀瀀攀爀椀洀攀渀琀愀氀 椀渀瘀攀猀琀椀最愀琀椀漀渀猀 漀昀 琀栀攀 爀愀搀椀愀琀椀漀渀 椀渀琀攀渀猀椀琀礀 愀猀 愀 昀甀渀挀琀椀漀渀 漀昀 琀攀洀瀀攀爀愀琀甀爀攀 愀渀搀 昀爀攀焀甀攀渀挀礀Ⰰ 刀吀⠀昀⤀ 愀戀漀瘀攀Ⰰ 最椀瘀攀 眀攀氀氀ⴀ搀攀昀椀渀攀搀 爀攀瀀爀漀搀甀挀椀戀氀攀 爀攀猀甀氀琀猀Ⰰ 渀漀琀 瘀攀爀礀 猀攀渀猀椀琀椀瘀攀 琀漀 琀栀攀 洀愀琀攀爀椀愀氀 漀昀 琀栀攀 漀瘀攀渀Ⰰ 攀琀挀⸀ 䘀漀爀 愀 猀洀愀氀氀 栀漀氀攀Ⰰ 琀栀攀 爀愀搀椀愀琀椀漀渀 攀洀椀琀琀攀搀 眀椀氀氀 戀攀 愀 爀攀瀀爀攀猀攀渀琀愀琀椀瘀攀 猀愀洀瀀氀攀 漀昀 琀栀攀 爀愀搀椀愀琀椀漀渀 戀漀甀渀挀椀渀最 愀爀漀甀渀搀 椀渀猀椀搀攀 琀栀攀 戀漀砀⸀  ਀㰀⼀戀爀㸀㰀戀爀㸀 In fact, the general shape of the function RT(f) was very reminiscent of the energy distribution in a gas in thermal equilibrium, as analyzed by Maxwell and Boltzmann. In contrast to the molecules in a gas, the different wavelengths of radiation do not collide with each other in the middle of the oven. Nevertheless, energy can shift from one mode to the other through the intermediary of the oscillators in the wall - which is to say, just the atoms and molecules the wall is made of. An electromagnetic wave in the oven can set a charge oscillating in the wall, energy can be transferred to other charges in the wall, which will in general oscillate and radiate at different frequencies. Thus the different frequencies of radiation inside the oven will come to thermal equilibrium.਀

਀ ਀㰀⼀瀀㸀 ਀ ਀ ਀
biography
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਀ ਀ ਀ 圀椀氀栀攀氀洀 圀椀攀渀 眀愀猀 戀漀爀渀 漀渀 䨀愀渀甀愀爀礀 ㄀㌀Ⰰ ㄀㠀㘀㐀 愀琀 䘀椀猀挀栀栀愀甀猀攀渀Ⰰ 椀渀 䔀愀猀琀 倀爀甀猀猀椀愀⸀ 䠀攀 眀愀猀 琀栀攀 猀漀渀 漀昀 琀栀攀 氀愀渀搀漀眀渀攀爀 䌀愀爀氀 圀椀攀渀Ⰰ 愀渀搀 猀攀攀洀攀搀 搀攀猀琀椀渀攀搀 昀漀爀 琀栀攀 氀椀昀攀 漀昀 愀 最攀渀琀氀攀洀愀渀 昀愀爀洀攀爀Ⰰ 戀甀琀 愀渀 攀挀漀渀漀洀椀挀 挀爀椀猀椀猀 愀渀搀 栀椀猀 漀眀渀 猀攀挀爀攀琀 猀攀渀猀攀 漀昀 瘀漀挀愀琀椀漀渀 氀攀搀 栀椀洀 琀漀 唀渀椀瘀攀爀猀椀琀礀 猀琀甀搀椀攀猀⸀ 圀栀攀渀 椀渀 ㄀㠀㘀㘀 栀椀猀 瀀愀爀攀渀琀猀 洀漀瘀攀搀 琀漀 䐀爀愀挀栀猀琀攀椀渀Ⰰ 椀渀 琀栀攀 刀愀猀琀攀渀戀甀爀最 搀椀猀琀爀椀挀琀 漀昀 䔀愀猀琀 倀爀甀猀猀椀愀Ⰰ 圀椀攀渀 眀攀渀琀 琀漀 猀挀栀漀漀氀 椀渀 ㄀㠀㜀㤀 昀椀爀猀琀 愀琀 刀愀猀琀攀渀戀甀爀最 愀渀搀 氀愀琀攀爀Ⰰ 昀爀漀洀 ㄀㠀㠀　 琀椀氀氀 ㄀㠀㠀㈀Ⰰ 愀琀 琀栀攀 䌀椀琀礀 匀挀栀漀漀氀 愀琀 䠀攀椀搀攀氀戀攀爀最⸀  ਀䄀昀琀攀爀 氀攀愀瘀椀渀最 猀挀栀漀漀氀 栀攀 眀攀渀琀Ⰰ 椀渀 ㄀㠀㠀㈀Ⰰ 琀漀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䜀琀琀椀渀最攀渀 琀漀 猀琀甀搀礀 洀愀琀栀攀洀愀琀椀挀猀 愀渀搀 琀栀攀 渀愀琀甀爀愀氀 猀挀椀攀渀挀攀猀 愀渀搀 椀渀 琀栀攀 猀愀洀攀 礀攀愀爀 愀氀猀漀 琀漀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䈀攀爀氀椀渀⸀ 䘀爀漀洀 ㄀㠀㠀㌀ 甀渀琀椀氀 ㄀㠀㠀㔀 栀攀 眀漀爀欀攀搀 椀渀 琀栀攀 氀愀戀漀爀愀琀漀爀礀 漀昀 䠀攀爀洀愀渀渀 瘀漀渀 䠀攀氀洀栀漀氀琀稀 愀渀搀 椀渀 ㄀㠀㠀㘀 栀攀 琀漀漀欀 栀椀猀 搀漀挀琀漀爀愀琀攀 眀椀琀栀 愀 琀栀攀猀椀猀 漀渀 栀椀猀 攀砀瀀攀爀椀洀攀渀琀猀 漀渀 琀栀攀 搀椀昀昀爀愀挀琀椀漀渀 漀昀 氀椀最栀琀 漀渀 猀攀挀琀椀漀渀猀 漀昀 洀攀琀愀氀猀 愀渀搀 漀渀 琀栀攀 椀渀昀氀甀攀渀挀攀 漀昀 洀愀琀攀爀椀愀氀猀 漀渀 琀栀攀 挀漀氀漀甀爀 漀昀 爀攀昀爀愀挀琀攀搀 氀椀最栀琀⸀ ਀䠀椀猀 猀琀甀搀椀攀猀 眀攀爀攀 琀栀攀渀 椀渀琀攀爀爀甀瀀琀攀搀 戀礀 琀栀攀 椀氀氀渀攀猀猀 漀昀 栀椀猀 昀愀琀栀攀爀 愀渀搀Ⰰ 甀渀琀椀氀 ㄀㠀㤀　Ⰰ 栀攀 栀攀氀瀀攀搀 椀渀 琀栀攀 洀愀渀愀最攀洀攀渀琀 漀昀 栀椀猀 昀愀琀栀攀爀✀猀 氀愀渀搀⸀ 䠀攀 眀愀猀Ⰰ 栀漀眀攀瘀攀爀Ⰰ 愀戀氀攀 琀漀 猀瀀攀渀搀Ⰰ 搀甀爀椀渀最 琀栀椀猀 瀀攀爀椀漀搀Ⰰ 漀渀攀 猀攀洀攀猀琀攀爀 眀椀琀栀 䠀攀氀洀栀漀氀琀稀 愀渀搀 椀渀 ㄀㠀㠀㜀 栀攀 搀椀搀 攀砀瀀攀爀椀洀攀渀琀猀 漀渀 琀栀攀 瀀攀爀洀攀愀戀椀氀椀琀礀 漀昀 洀攀琀愀氀猀 琀漀 氀椀最栀琀 愀渀搀 栀攀愀琀 爀愀礀猀⸀ 圀栀攀渀 栀椀猀 昀愀琀栀攀爀✀猀 氀愀渀搀 眀愀猀 猀漀氀搀 栀攀 爀攀琀甀爀渀攀搀 琀漀 琀栀攀 氀愀戀漀爀愀琀漀爀礀 漀昀 䠀攀氀洀栀漀氀琀稀Ⰰ 眀栀漀 栀愀搀 戀攀攀渀 洀漀瘀攀搀 琀漀Ⰰ 愀渀搀 栀愀搀 戀攀挀漀洀攀 倀爀攀猀椀搀攀渀琀 漀昀Ⰰ 琀栀攀 倀栀礀猀椀欀愀氀椀猀挀栀ⴀ吀攀挀栀渀椀猀挀栀攀 刀攀椀挀栀猀愀渀猀琀愀氀琀Ⰰ 攀猀琀愀戀氀椀猀栀攀搀 昀漀爀 琀栀攀 猀琀甀搀礀 漀昀 椀渀搀甀猀琀爀椀愀氀 瀀爀漀戀氀攀洀猀⸀ 䠀攀爀攀 栀攀 爀攀洀愀椀渀攀搀 甀渀琀椀氀 ㄀㠀㤀㘀 眀栀攀渀 栀攀 眀愀猀 愀瀀瀀漀椀渀琀攀搀 倀爀漀昀攀猀猀漀爀 漀昀 倀栀礀猀椀挀猀 愀琀 䄀椀砀ⴀ氀愀ⴀ䌀栀愀瀀攀氀氀攀 椀渀 猀甀挀挀攀猀猀椀漀渀 琀漀 倀栀椀氀椀瀀瀀 䰀攀渀愀爀搀⸀ 䤀渀 ㄀㠀㤀㤀Ⰰ 栀攀 眀愀猀 愀瀀瀀漀椀渀琀攀搀 倀爀漀昀攀猀猀漀爀 漀昀 倀栀礀猀椀挀猀 愀琀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䜀椀攀猀猀攀渀⸀ 䤀渀 ㄀㤀　　 栀攀 戀攀挀愀洀攀 倀爀漀昀攀猀猀漀爀 漀昀 琀栀攀 猀愀洀攀 猀甀戀樀攀挀琀 愀琀 圀ﰀ爀稀戀甀爀最Ⰰ 椀渀 猀甀挀挀攀猀猀椀漀渀 琀漀 圀⸀䌀⸀ 刀渀琀最攀渀Ⰰ 愀渀搀 椀渀 琀栀椀猀 礀攀愀爀 栀攀 瀀甀戀氀椀猀栀攀搀 栀椀猀 䰀攀栀爀戀甀挀栀 搀攀爀 䠀礀搀爀漀搀礀渀愀洀椀欀 ⠀吀攀砀琀戀漀漀欀 漀昀 栀礀搀爀漀搀礀渀愀洀椀挀猀⤀⸀ ਀㰀栀㐀㸀䤀渀 ㄀㤀　㈀ 栀攀 眀愀猀 椀渀瘀椀琀攀搀 琀漀 猀甀挀挀攀攀搀 䰀甀搀眀椀最 䈀漀氀琀稀洀愀渀渀 愀猀 倀爀漀昀攀猀猀漀爀 漀昀 倀栀礀猀椀挀猀 愀琀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䰀攀椀瀀稀椀最 愀渀搀 椀渀 ㄀㤀　㘀 琀漀 猀甀挀挀攀攀搀 䐀爀甀搀攀 愀猀 倀爀漀昀攀猀猀漀爀 漀昀 倀栀礀猀椀挀猀 愀琀 琀栀攀 唀渀椀瘀攀爀猀椀琀礀 漀昀 䈀攀爀氀椀渀㬀 戀甀琀 栀攀 爀攀昀甀猀攀搀 戀漀琀栀 琀栀攀猀攀 椀渀瘀椀琀愀琀椀漀渀猀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 ㄀㤀㈀　 栀攀 眀愀猀 愀瀀瀀漀椀渀琀攀搀 倀爀漀昀攀猀猀漀爀 漀昀 倀栀礀猀椀挀猀 愀琀 䴀甀渀椀挀栀Ⰰ 眀栀攀爀攀 栀攀 爀攀洀愀椀渀攀搀 琀栀爀漀甀最栀漀甀琀 琀栀攀 爀攀猀琀 漀昀 栀椀猀 氀椀昀攀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 愀搀搀椀琀椀漀渀 琀漀 琀栀攀 攀愀爀氀礀 眀漀爀欀 愀氀爀攀愀搀礀 洀攀渀琀椀漀渀攀搀Ⰰ 圀椀攀渀 眀漀爀欀攀搀Ⰰ 愀琀 琀栀攀 倀栀礀猀椀欀愀氀椀猀挀栀ⴀ吀攀挀栀渀椀猀挀栀攀 刀攀椀挀栀猀愀渀猀琀愀氀琀Ⰰ 眀椀琀栀 䠀漀氀戀漀爀渀 漀渀 洀攀琀栀漀搀猀 漀昀 洀攀愀猀甀爀椀渀最 栀椀最栀 琀攀洀瀀攀爀愀琀甀爀攀猀 眀椀琀栀 琀栀攀 䰀攀 䌀栀愀琀攀氀椀攀爀 琀栀攀爀洀漀攀氀攀洀攀渀琀猀 愀渀搀 愀琀 琀栀攀 猀愀洀攀 琀椀洀攀 搀椀搀 琀栀攀漀爀攀琀椀挀愀氀 眀漀爀欀 漀渀 琀栀攀爀洀漀搀礀渀愀洀椀挀猀Ⰰ 攀猀瀀攀挀椀愀氀氀礀 漀渀 琀栀攀 氀愀眀猀 最漀瘀攀爀渀椀渀最 琀栀攀 爀愀搀椀愀琀椀漀渀 漀昀 栀攀愀琀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 ㄀㠀㤀㌀ 栀攀 愀渀渀漀甀渀挀攀搀 琀栀攀 氀愀眀 眀栀椀挀栀 猀琀愀琀攀猀 琀栀愀琀 琀栀攀 眀愀瘀攀氀攀渀最琀栀 挀栀愀渀最攀猀 眀椀琀栀 琀栀攀 琀攀洀瀀攀爀愀琀甀爀攀Ⰰ 愀 氀愀眀 眀栀椀挀栀 氀愀琀攀爀 戀攀挀愀洀攀 琀栀攀 氀愀眀 漀昀 搀椀猀瀀氀愀挀攀洀攀渀琀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 ㄀㠀㤀㐀 栀攀 瀀甀戀氀椀猀栀攀搀 愀 瀀愀瀀攀爀 漀渀 琀攀洀瀀攀爀愀琀甀爀攀 愀渀搀 琀栀攀 攀渀琀爀漀瀀礀 漀昀 爀愀搀椀愀琀椀漀渀Ⰰ 椀渀 眀栀椀挀栀 琀栀攀 琀攀爀洀猀 琀攀洀瀀攀爀愀琀甀爀攀 愀渀搀 攀渀琀爀漀瀀礀 眀攀爀攀 攀砀琀攀渀搀攀搀 琀漀 爀愀搀椀愀琀椀漀渀 椀渀 攀洀瀀琀礀 猀瀀愀挀攀⸀ 䤀渀 琀栀椀猀 眀漀爀欀 栀攀 眀愀猀 氀攀搀 琀漀 搀攀昀椀渀攀 愀渀 椀搀攀愀氀 戀漀搀礀Ⰰ 眀栀椀挀栀 栀攀 挀愀氀氀攀搀 琀栀攀 戀氀愀挀欀 戀漀搀礀Ⰰ 眀栀椀挀栀 挀漀洀瀀氀攀琀攀氀礀 愀戀猀漀爀戀猀 愀氀氀 爀愀搀椀愀琀椀漀渀猀⸀ 䤀渀 ㄀㠀㤀㘀 栀攀 瀀甀戀氀椀猀栀攀搀 琀栀攀 昀漀爀洀甀氀愀 漀昀 圀椀攀渀Ⰰ 眀栀椀挀栀 眀愀猀 琀栀攀 爀攀猀甀氀琀 漀昀 眀漀爀欀 甀渀搀攀爀琀愀欀攀渀 琀漀 昀椀渀搀 愀 昀漀爀洀甀氀愀 昀漀爀 琀栀攀 挀漀洀瀀漀猀椀琀椀漀渀 漀昀 琀栀攀 爀愀搀椀愀琀椀漀渀 漀昀 猀甀挀栀 愀 戀氀愀挀欀 戀漀搀礀⸀ 䰀愀琀攀爀 椀琀 眀愀猀 瀀爀漀瘀攀搀 琀栀愀琀 琀栀椀猀 昀漀爀洀甀氀愀 椀猀 瘀愀氀椀搀 漀渀氀礀 昀漀爀 琀栀攀 猀栀漀爀琀 眀愀瘀攀猀Ⰰ 戀甀琀 圀椀攀渀✀猀 眀漀爀欀 攀渀愀戀氀攀搀 䴀愀砀 倀氀愀渀挀欀 琀漀 爀攀猀漀氀瘀攀 琀栀攀 瀀爀漀戀氀攀洀 漀昀 爀愀搀椀愀琀椀漀渀 椀渀 琀栀攀爀洀愀氀 攀焀甀椀氀椀戀爀椀甀洀 戀礀 洀攀愀渀猀 漀昀 焀甀愀渀琀甀洀 瀀栀礀猀椀挀猀⸀ 䘀漀爀 琀栀椀猀 眀漀爀欀 圀椀攀渀 眀愀猀 愀眀愀爀搀攀搀 琀栀攀 一漀戀攀氀 倀爀椀稀攀 昀漀爀 倀栀礀猀椀挀猀 昀漀爀 ㄀㤀㄀㄀⸀ 䄀渀 椀渀琀攀爀攀猀琀椀渀最 瀀漀椀渀琀 愀戀漀甀琀 椀琀 椀猀 琀栀愀琀 琀栀椀猀 琀栀攀漀爀攀琀椀挀愀氀 眀漀爀欀 挀愀洀攀 昀爀漀洀 愀渀 䤀渀猀琀椀琀甀琀攀 搀攀瘀漀琀攀搀 琀漀 琀攀挀栀渀椀挀愀氀 瀀爀漀戀氀攀洀猀 愀渀搀 椀琀 氀攀搀 琀漀 渀攀眀 琀攀挀栀渀椀焀甀攀猀 昀漀爀 椀氀氀甀洀椀渀愀琀椀漀渀 愀渀搀 琀栀攀 洀攀愀猀甀爀攀洀攀渀琀 漀昀 栀椀最栀 琀攀洀瀀攀爀愀琀甀爀攀猀⸀ ਀圀栀攀渀 圀椀攀渀 洀漀瘀攀搀Ⰰ 椀渀 ㄀㠀㤀㘀Ⰰ 琀漀 䄀椀砀ⴀ氀愀ⴀ䌀栀愀瀀攀氀氀攀 琀漀 猀甀挀挀攀攀搀 䰀攀渀愀爀搀Ⰰ 栀攀 昀漀甀渀搀 琀栀攀爀攀 愀 氀愀戀漀爀愀琀漀爀礀 攀焀甀椀瀀瀀攀搀 昀漀爀 琀栀攀 猀琀甀搀礀 漀昀 攀氀攀挀琀爀椀挀愀氀 搀椀猀挀栀愀爀最攀猀 椀渀 瘀愀挀甀漀 愀渀搀 椀渀 ㄀㠀㤀㜀 栀攀 戀攀最愀渀 琀漀 眀漀爀欀 漀渀 琀栀攀 渀愀琀甀爀攀 漀昀 挀愀琀栀漀搀攀 爀愀礀猀⸀ 唀猀椀渀最 愀 瘀攀爀礀 栀椀最栀 瘀愀挀甀甀洀 琀甀戀攀 眀椀琀栀 愀 䰀攀渀愀爀搀 眀椀渀搀漀眀Ⰰ 栀攀 挀漀渀昀椀爀洀攀搀 琀栀攀 搀椀猀挀漀瘀攀爀礀 琀栀愀琀 搀攀愀渀 倀攀爀爀椀渀 栀愀搀 洀愀搀攀 琀眀漀 礀攀愀爀猀 攀愀爀氀椀攀爀Ⰰ 琀栀愀琀 挀愀琀栀漀搀攀 爀愀礀猀 愀爀攀 挀漀洀瀀漀猀攀搀 漀昀 爀愀瀀椀搀氀礀ⴀ洀漀瘀椀渀最Ⰰ 渀攀最愀琀椀瘀攀氀礀ⴀ挀栀愀爀最攀搀 瀀愀爀琀椀挀氀攀猀 ⠀攀氀攀挀琀爀漀渀猀⤀⸀ 䄀渀搀 琀栀攀渀Ⰰ 愀氀洀漀猀琀 愀琀 琀栀攀 猀愀洀攀 琀椀洀攀 愀猀 匀椀爀 䨀⸀䨀⸀ 吀栀漀洀猀漀渀 椀渀 䌀愀洀戀爀椀搀最攀Ⰰ 戀甀琀 戀礀 愀 搀椀昀昀攀爀攀渀琀 洀攀琀栀漀搀Ⰰ 栀攀 洀攀愀猀甀爀攀搀 琀栀攀 爀攀氀愀琀椀漀渀 漀昀 琀栀攀 攀氀攀挀琀爀椀挀 挀栀愀爀最攀 漀渀 琀栀攀猀攀 瀀愀爀琀椀挀氀攀猀 琀漀 琀栀攀椀爀 洀愀猀猀 愀渀搀 昀漀甀渀搀Ⰰ 愀猀 吀栀漀洀猀漀渀 搀椀搀Ⰰ 琀栀愀琀 琀栀攀礀 愀爀攀 愀戀漀甀琀 琀眀漀 琀栀漀甀猀愀渀搀 琀椀洀攀猀 氀椀最栀琀攀爀 琀栀愀渀 琀栀攀 愀琀漀洀猀 漀昀 栀礀搀爀漀最攀渀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 ㄀㠀㤀㠀 圀椀攀渀 猀琀甀搀椀攀搀 琀栀攀 挀愀渀愀氀 爀愀礀猀 搀椀猀挀漀瘀攀爀攀搀 戀礀 䜀漀氀搀猀琀攀椀渀 愀渀搀 挀漀渀挀氀甀搀攀搀 琀栀愀琀 琀栀攀礀 眀攀爀攀 琀栀攀 瀀漀猀椀琀椀瘀攀 攀焀甀椀瘀愀氀攀渀琀 漀昀 琀栀攀 渀攀最愀琀椀瘀攀氀礀ⴀ挀栀愀爀最攀搀 挀愀琀栀漀搀攀 爀愀礀猀⸀ 䠀攀 洀攀愀猀甀爀攀搀 琀栀攀椀爀 搀攀瘀椀愀琀椀漀渀 戀礀 洀愀最渀攀琀椀挀 愀渀搀 攀氀攀挀琀爀椀挀 昀椀攀氀搀猀 愀渀搀 挀漀渀挀氀甀搀攀搀 琀栀愀琀 琀栀攀礀 愀爀攀 挀漀洀瀀漀猀攀搀 漀昀 瀀漀猀椀琀椀瘀攀氀礀ⴀ挀栀愀爀最攀搀 瀀愀爀琀椀挀氀攀猀 渀攀瘀攀爀 栀攀愀瘀椀攀爀 琀栀愀渀 攀氀攀挀琀爀漀渀猀⸀ ਀吀栀攀 洀攀琀栀漀搀 甀猀攀搀 戀礀 圀椀攀渀 爀攀猀甀氀琀攀搀 猀漀洀攀 ㈀　 礀攀愀爀猀 氀愀琀攀爀 椀渀 琀栀攀 猀瀀攀挀琀爀漀最爀愀瀀栀礀 漀昀 洀愀猀猀攀猀Ⰰ 眀栀椀挀栀 栀愀猀 洀愀搀攀 瀀漀猀猀椀戀氀攀 琀栀攀 瀀爀攀挀椀猀攀 洀攀愀猀甀爀攀洀攀渀琀 漀昀 琀栀攀 洀愀猀猀攀猀 漀昀 瘀愀爀椀漀甀猀 愀琀漀洀猀 愀渀搀 琀栀攀椀爀 椀猀漀琀漀瀀攀猀Ⰰ 渀攀挀攀猀猀愀爀礀 昀漀爀 琀栀攀 挀愀氀挀甀氀愀琀椀漀渀 漀昀 琀栀攀 攀渀攀爀最椀攀猀 爀攀氀攀愀猀攀搀 戀礀 渀甀挀氀攀愀爀 爀攀愀挀琀椀漀渀猀⸀ 䤀渀 ㄀㤀　　 圀椀攀渀 瀀甀戀氀椀猀栀攀搀 愀 琀栀攀漀爀攀琀椀挀愀氀 瀀愀瀀攀爀 漀渀 琀栀攀 瀀漀猀猀椀戀椀氀椀琀礀 漀昀 愀渀 攀氀攀挀琀爀漀洀愀最渀攀琀椀挀 戀愀猀椀猀 昀漀爀 洀攀挀栀愀渀椀挀猀⸀ 匀甀戀猀攀焀甀攀渀琀氀礀 栀攀 搀椀搀 昀甀爀琀栀攀爀 眀漀爀欀 漀渀 琀栀攀 挀愀渀愀氀 爀愀礀猀Ⰰ 猀栀漀眀椀渀最Ⰰ 椀渀 ㄀㤀㄀㈀Ⰰ 琀栀愀琀Ⰰ 椀昀 琀栀攀 瀀爀攀猀猀甀爀攀 椀猀 渀漀琀 攀砀琀爀攀洀攀氀礀 眀攀愀欀Ⰰ 琀栀攀猀攀 爀愀礀猀 氀漀猀攀 愀渀搀 爀攀最愀椀渀Ⰰ 戀礀 挀漀氀氀椀猀椀漀渀 眀椀琀栀 愀琀漀洀猀 漀昀 爀攀猀椀搀甀愀氀 最愀猀Ⰰ 琀栀攀椀爀 攀氀攀挀琀爀椀挀 挀栀愀爀最攀 愀氀漀渀最 琀栀攀椀爀 挀漀甀爀猀攀 漀昀 琀爀愀瘀攀氀⸀ 䤀渀 ㄀㤀㄀㠀 栀攀 瀀甀戀氀椀猀栀攀搀 昀甀爀琀栀攀爀 眀漀爀欀 漀渀 琀栀攀猀攀 爀愀礀猀 漀渀 琀栀攀 洀攀愀猀甀爀攀洀攀渀琀 漀昀 琀栀攀 瀀爀漀最爀攀猀猀椀瘀攀 搀攀挀爀攀愀猀攀 漀昀 琀栀攀椀爀 氀甀洀椀渀漀猀椀琀礀 愀昀琀攀爀 琀栀攀礀 氀攀愀瘀攀 琀栀攀 挀愀琀栀漀搀攀 愀渀搀 昀爀漀洀 琀栀攀猀攀 攀砀瀀攀爀椀洀攀渀琀猀 栀攀 搀攀搀甀挀攀搀 眀栀愀琀 挀氀愀猀猀椀挀愀氀 瀀栀礀猀椀挀猀 挀愀氀氀猀 琀栀攀 搀攀挀愀礀 漀昀 琀栀攀 氀甀洀椀渀漀甀猀 瘀椀戀爀愀琀椀漀渀猀 椀渀 琀栀攀 愀琀漀洀猀Ⰰ 眀栀椀挀栀 挀漀爀爀攀猀瀀漀渀搀猀 椀渀 焀甀愀渀琀甀洀 瀀栀礀猀椀挀猀 琀漀 琀栀攀 氀椀洀椀琀攀搀 搀甀爀愀琀椀漀渀 漀昀 攀砀挀椀琀攀搀 猀琀愀琀攀猀 漀昀 愀琀漀洀猀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 琀栀椀猀Ⰰ 愀渀搀 漀琀栀攀爀Ⰰ 爀攀猀瀀攀挀琀猀 圀椀攀渀✀猀 眀漀爀欀 挀漀渀琀爀椀戀甀琀攀搀 琀漀 琀栀攀 琀爀愀渀猀椀琀椀漀渀 昀爀漀洀 一攀眀琀漀渀椀愀渀 琀漀 焀甀愀渀琀甀洀 瀀栀礀猀椀挀猀⸀ 䄀猀 䴀愀砀 瘀漀渀 䰀愀甀攀 眀爀漀琀攀 漀昀 栀椀洀Ⰰ 栀椀猀 ∀椀洀洀漀爀琀愀氀 最氀漀爀礀∀ 眀愀猀 琀栀愀琀 ∀栀攀 氀攀搀 甀猀 琀漀 琀栀攀 瘀攀爀礀 最愀琀攀猀 漀昀 焀甀愀渀琀甀洀 瀀栀礀猀椀挀猀∀⸀ ਀圀椀攀渀 眀愀猀 愀 洀攀洀戀攀爀 漀昀 琀栀攀 䄀挀愀搀攀洀椀攀猀 漀昀 匀挀椀攀渀挀攀猀 漀昀 䈀攀爀氀椀渀Ⰰ 䜀琀琀椀渀最攀渀Ⰰ 嘀椀攀渀渀愀Ⰰ 匀琀漀挀欀栀漀氀洀Ⰰ 䌀栀爀椀猀琀椀愀渀椀愀 愀渀搀 圀愀猀栀椀渀最琀漀渀Ⰰ 愀渀搀 愀渀 䠀漀渀漀爀愀爀礀 洀攀洀戀攀爀 漀昀 琀栀攀 倀栀礀猀椀挀愀氀 匀漀挀椀攀琀礀 漀昀 䘀爀愀渀欀昀甀爀琀ⴀ漀渀ⴀ䴀愀椀渀⸀ ਀㰀⼀栀㐀㸀㰀栀㐀㸀䤀渀 ㄀㠀㤀㠀 栀攀 洀愀爀爀椀攀搀 䰀甀椀猀攀 䴀攀栀氀攀爀 漀昀 䄀椀砀ⴀ氀愀ⴀ䌀栀愀瀀攀氀氀攀⸀ 吀栀攀礀 栀愀搀 昀漀甀爀 挀栀椀氀搀爀攀渀⸀ 䠀攀 搀椀攀搀 椀渀 䴀甀渀椀挀栀 漀渀 䄀甀最甀猀琀 ㌀　Ⰰ ㄀㤀㈀㠀⸀ ਀ ਀ ਀ ਀ ਀㰀⼀栀㐀㸀 ਀ ਀ ਀ ਀ sponsors link òþíèíã àâòî
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